A Unification of Two Refinements of Euler's Partition Theorem

TL;DR

This paper unifies two refinements of Euler's partition theorem using combinatorial methods, specifically Bessenrodt's insertion algorithm, to generalize the Andrews-Olsson partition identity.

Contribution

It provides a unified combinatorial framework for two existing refinements of Euler's partition theorem, extending the Andrews-Olsson identity.

Findings

Unified the two refinements of Euler's partition theorem.

Extended the Andrews-Olsson partition identity.

Applied Bessenrodt's insertion algorithm in a new context.

Abstract

We obtain a unification of two refinements of Euler's partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt's insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction.